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Headphone gestures not working on Linux
(yiffit.net)
A place for AUDIO enthusiast to share, discuss and listen to others people setups
Most likely the headphones are doing the translation and end up sending a normal signal like any other headphone would. Meaning that if it's not working it's either the app you are using in Linux, the drivers, or the DE.
Give us some more details on what software you were using on Linux, what distro you are using and what DE
Maybe start by checking if other headphones have the same issue and maybe the arch wiki since it has a Bluetooth page